Use the FTC to find .
We have f(x) = 3x2, a = 0, and b = 2. We know the function F(x) = x3 is an antiderivative of f. Starting with the FTC and plugging in the appropriate values and functions, we get
Be warned - even if the lower limit of integration is zero, F(0) might not be zero.
An antiderivative of f(t) = -sin t is F(t) = cos t. We have a = 0 and .
Plugging things into the FTC, we get
When evaluating integrals by hand, make sure you leave the answers in exact form.
ex is its own derivative, and therefore its own antiderivative.
This is approximately equal to 1.718, but you should give e1 – 1 as your final answer, because e1 – 1 is exact and 1.718 is only an approximation.
When the integrand has multiple terms, be careful to keep track of signs correctly. As you work out F(b) and F(a), keep parentheses around them so the signs don't get mixed up.
An antiderivative of f(x) = (x2 – x) is . Using the FTC,
We leave the answer as because this answer is exact.
An antiderivative of 2x + 4x3 is x2 + x4. Using the FTC and the shortcut notation,
We expand the right-hand side and simplify:
Make it rain.